Theorem int0 4904

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4438	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3433	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4896	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 231	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 264	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2733	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ∅c0 4273  ∩ cint 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-dif 3892  df-nul 4274  df-int 4890

This theorem is referenced by:  unissint  4914  uniintsn  4927  rint0  4930  intex  5285  intnex  5286  oev2  8458  fiint  9237  elfi2  9327  fi0  9333  cardmin2  9923  00lsp  20976  cmpfi  23373  ptbasfi  23546  fbssint  23803  fclscmp  23995  zarcmplem  34025  rankeq1o  36353  bj-0int  37413  heibor1lem  38130  ipoglb0  49469  mreclat  49472